1. Field of the Invention
The present invention relates to methods for integrating satellite and inertial navigation systems. More particularly, this invention pertains to a method for checking the reliability of nominal position findings.
2. Description of the Prior Art
Position and velocity with respect to a given coordinate system are the essential outputs of a navigation system. Inertial navigation systems supply these variables at a high update rate and can be jammed only with difficulty as they require solely internal sensors and no external system components. Their disadvantage resides in the degradation of their accuracies over time. In contrast, nominal (e.g. satellite) navigation systems achieve excellent long-term accuracy, yet possess only a limited update rate while being highly susceptible to jamming as they are based upon the principle of measuring the propagation time of an electromagnetic wave between a satellite and a receiver.
The disadvantages of each of the above systems can be eliminated by their integration or combination. Error variables are determined from the output variables of the two systems and converted by an optimizing filter (generally a Kalman filter) into so-called support variables. The support variables are employed for continual correction of the inertial system.
As satellite navigation systems are highly susceptible to jamming, it is essential for safety-critical applications to employ a means for error identification that checks the output variables of the satellite navigation system for consistency and decides whether they may be used for support.
To use satellite navigation for position finding and determining velocity, it is necessary to measure and evaluate at least four satellite signals. Should an erroneous measurement be identified, the number of required measured values is increased to at least five. In practice, however, six or more measurements are often employed. The measured values are processed in a mathematically appropriate manner and are then subjected to a hypothesis test. A statistical statement can be derived from the test as to whether one or more measurements is erroneous.
The statistical characteristics of the hypothesis test improve, as a step function, with the number of available measured values. Due to the mathematical character of the use of satellite navigation for position findings, findings determined in other ways can be included in the hypothesis test to increase its statistical reliability.
Statistical methods are employed for error identification in satellite navigation and can be split into two classes:
1. Methods based upon individual measurements; and PA1 2. Methods based upon series of measurements.
All methods based upon individual measurements are subject to an assumption that statistically independent, noisy, possible erroneous measurements exist at the respective measurement time. The basic measurement equation, which produces the relationship between (n&gt;4) available measurements and the receiver position and velocity, is described by the following linear, overdefined equation system: EQU V=G.cndot.X.sub.act +.epsilon.
where y, a vector of dimension n.times.1, contains the difference between the measured range (pseudo-range) and that estimated on the basis of the nominal state vector; X.sub.act describes the error between the nominal value and the four-dimensional state vector (position and/or velocity, and timebase offset) calculated from the current measurement; G represents the linearized n.times.4 measurement matrix; and .epsilon., a vector of dimension n.times.1, describes the measurement noise, including any measurement errors.
The least squares method is employed, infra, to illustrate the group of methods based on individual measurements. All other methods of this kind can be related back to it or to work on similar mathematical procedures.
A solution for x is first calculated in accordance with the least squares method. EQU X.sub.LS =(G.sup.T G).sup.-1 G.sup.T y
This is employed for determining the expected value for the measurement factor y. EQU y=G.cndot.X.sub.LS
The expected value is subtracted from the actual measurement vector, y, to produce a residue vector w. EQU w:=y-y=(1-G(G.sup.T G).sup.-1 G.sup.T).epsilon.
The residue vector w is multiplied by its transposed vector to form a scalar test variable T: EQU T:=W.sup.T W
It is known that measurement errors .epsilon..sub.i of Gaussian distribution produce a test variable T having a Chi-square distribution with n-4 degrees of freedom. The test variable T can thus be subjected to a Chi-squared hypothesis test and a statistical statement can be made about the expected state vector inaccuracy. Should the numerical value of the inaccuracy exceed a predetermined limit S, it is then assumed that the measurement is erroneous.
It is known to include the barometric altitude P in the measurement equation in avionics applications. For this, the measurement matrix G, containing the direction cosine between the receiver and the n satellites, is expanded by one line: ##EQU1##
where .sigma..sub.sat /.sigma..sub.b is the ratio of the noise in the pseudo-range measurements to that from the barometric altimeter.
The residue vector dimension is likewise increased: EQU W'=(y1-y1,y2-y2, . . . ,Yn-yn,(yZ.sub.b -yZ) .div.(.sigma..sub.b /.sigma..sub.sat)).sup.T,
where YZ.sub.b is the difference between the barometric altitude and the nominal altitude and yz is the altitude error calculated using least squares.
The above measure increases the degrees of freedom of the Chi-square distribution of T. This results in improvement of the statistical characteristics of the hypothesis test for the same number of received satellites. An additional important advantage is that test feasibility is insured, even for the case of n=4 usable satellites, since the dimension of the new measurement matrix G' is equal to n+1.
Error identification methods based on series of measurements are based on an algorithm published as early as 1965 under the name "Adaptive Kalman Filtering" or "Multiple Model Estimation Algorithm (MMEA)".
The MMEA employs a bank of Kalman filters in parallel. Each Kalman filter models all the relevant parameters of the system to be monitored as well as an additional parameter describing a hypothetical error. In the current field of application, a Kalman filter is employed for each received satellite, with the assumption made that the pseudo-range measurement is corrupted by a possible error that is a linear function of time. After a sufficiently long observation time, a statistical statement can be made as to whether one of the observed satellites is, in fact, erroneous.
U.S. Pat. No. 5,583,774 discloses a method for checking the reliability of nominal position findings on the basis of propagation time measurements. As taught by that patent, the measurement redundancy of the measured values available for error identification is increased by inertial systems. An algorithm is employed that depends on an observation time, preferably of 30 minutes.
The disadvantages of these methods are primarily the observation time required (measurement time period .gtoreq.30 minutes) and the high degree of complexity required for implementation.